Evaluating Branching Heuristics in Interval Constraint Propagation for Satisfiability

  • Calvin Huang
  • Soonho Kong
  • Sicun Gao
  • Damien ZuffereyEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11652)


Interval Constraint Propagation (ICP) is a powerful method for solving general nonlinear constraints over real numbers. ICP uses interval arithmetic to prune the space of potential solutions and, when the constraint propagation fails, divides the space into smaller regions and continues recursively. The original goal is to find paving boxes of all solutions to a problem. Already when the whole domain needs to be considered, branching methods do matter much. However, recent applications of ICP in decision procedures over the reals need only a single solution. Consequently, variable ordering in branching operations becomes even more important.

In this work, we compare three different branching heuristics for ICP. The first method, most commonly used, splits the problem in the dimension with the largest lower and upper bound. The two other types of branching methods try to exploit an integration of analytical/numerical properties of real functions and search-based methods. The second method, called smearing, uses gradient information of constraints to choose variables that have the highest local impact on pruning. The third method, lookahead branching, designs a measure function to compare the effect of all variables on pruning operations in the next several steps.

We evaluate the performance of our methods on over 11,000 benchmarks from various sources. While the different branching methods exhibit significant differences on larger instance, none is consistently better. This shows the need for further research on branching heuristics when ICP is used to find an unique solution rather than all solutions.



We thank the reviewers for their feedback and comments which helped us improve the paper. Calvin Huang was studying at MIT when that work was done. Damien Zufferey was funded in part by the DFG under Grant Agreement 389792660-TRR 248 and by the ERC under the Grant Agreement 610150.


  1. 1.
    Barrett, C., Fontaine, P., Tinelli, C.: The Satisfiability Modulo Theories Library (SMT-LIB) (2019).
  2. 2.
    Batnini, H., Michel, C., Rueher, M.: Mind the gaps: a new splitting strategy for consistency techniques. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 77–91. Springer, Heidelberg (2005). Scholar
  3. 3.
    Benhamou, F., Granvilliers, L.: Continuous and interval constraints. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, Chap. 16. Elsevier (2006)CrossRefGoogle Scholar
  4. 4.
    Biere, A., Fröhlich, A.: Evaluating CDCL variable scoring schemes. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 405–422. Springer, Cham (2015). Scholar
  5. 5.
    Csendes, T., Ratz, D.: Subdivision direction selection in interval methods for global optimization. SIAM J. Numer. Anal. 34(3), 922–938 (1997). Scholar
  6. 6.
    Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex Boolean structure. JSAT 1(3–4), 209–236 (2007)zbMATHGoogle Scholar
  7. 7.
    Gao, S., Avigad, J., Clarke, E.M.: \(\delta \)-complete decision procedures for satisfiability over the reals. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 286–300. Springer, Heidelberg (2012). Scholar
  8. 8.
    Gao, S., Kong, S., Clarke, E.M.: dReal: an SMT solver for nonlinear theories over the reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 208–214. Springer, Heidelberg (2013). Scholar
  9. 9.
    Gao, S., Kong, S., Clarke, E.M.: Proof generation from detla-decisions. In: Winkler, F., et al. (eds.) 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2014, Timisoara, Romania, 22–25 September 2014, pp. 156–163. IEEE Computer Society (2014).
  10. 10.
    Granvilliers, L.: Adaptive bisection of numerical CSPs. In: Milano, M. (ed.) CP 2012. LNCS, pp. 290–298. Springer, Heidelberg (2012). Scholar
  11. 11.
    Hales, T.C., et al.: A formal proof of the Kepler conjecture. Forum Math. Pi 5, e2 (2017). Scholar
  12. 12.
    Hamadi, Y., Ringwelski, G.: Boosting distributed constraint satisfaction. J. Heuristics 17(3), 251–279 (2011). Scholar
  13. 13.
    Kearfott, R.B., Novoa, M.: Algorithm 681: INTBIS, a portable interval Newton/bisection package. ACM Trans. Math. Softw. 16(2), 152–157 (1990). Scholar
  14. 14.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Design Automation Conference, pp. 530–535 (2001).
  15. 15.
    Purdom, P.W., Brown, C.A., Robertson, E.L.: Backtracking with multi-level dynamic search rearrangement. Acta Inf. 15(2), 99–113 (1981). Scholar
  16. 16.
    Reyes, V., Araya, I.: Probing-based variable selection heuristics for NCSPs. In: 2014 IEEE 26th International Conference on Tools with Artificial Intelligence, pp. 16–23, November 2014.
  17. 17.
    Rossi, F., van Beek, P., Walsh, T. (eds.): Handbook of Constraint Programming. Elsevier, New York (2006)zbMATHGoogle Scholar
  18. 18.
    Snyman, J.: Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms. Springer, New York (2005). Scholar
  19. 19.
    Stanley-Marbell, P., Francese, P.A., Rinard, M.: Encoder logic for reducing serial I/O power in sensors and sensor hubs. In: 2016 IEEE Hot Chips 28 Symposium (HCS), Cupertino, CA, USA, 21–23 August 2016, pp. 1–2. IEEE (2016).
  20. 20.
    Weihrauch, K.: Computable Analysis: An Introduction. Texts in Theoretical Computer Science. Springer, Heidelberg (2000). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Scale Labs Inc.San FranciscoUSA
  2. 2.Toyota Research InstituteCambridgeUSA
  3. 3.University of CaliforniaSan DiegoUSA
  4. 4.Max Planck Institute for Software SystemsKaiserslauternGermany

Personalised recommendations